Question: Simplify the following expression and state the condition under which the simplification is valid. $a = \dfrac{k^2 - 49}{k - 7}$
First factor the polynomial in the numerator. The numerator is in the form ${a^2} - {b^2}$ , which is a difference of two squares so we can factor it as $({a} + {b})({a} - {b})$ $ a = k$ $ b = \sqrt{49} = -7$ So we can rewrite the expression as: $a = \dfrac{({k} {-7})({k} + {7})} {k - 7} $ We can divide the numerator and denominator by $(k - 7)$ on condition that $k \neq 7$ Therefore $a = k + 7; k \neq 7$